Page:The Meaning of Relativity - Albert Einstein (1922).djvu/92

80 is also a vector. Since this is the case for an arbitrary choice of the $$dx_\sigma$$, it follows that

is a tensor, which we designate as the co-variant derivative of the tensor of the first rank (vector). Contracting this tensor, we obtain the divergence of the contra-variant tensor $$A^\mu$$. In this we must observe that according to (70),

If we put, further,

a quantity designated by Weyl as the contra-variant tensor density of the first rank, it follows that,

is a scalar density.

We get the law of parallel displacement for the co-variant vector $$B_\mu$$ by stipulating that the parallel displacement shall be effected in such a way that the scalar

remains unchanged, and that therefore